in the definition of a fiber bundle associated to a principal bundle,we have this: (P*F)~ s.t. (p,f)~(p.g,g-1.f)
where P is principal bundle and F is a Manifold. such way of gluing the points of P*F together gives us a quotient topological space such that satisfies in conditions of a fiber bundle with fiber F and a certain set of transition functions that comes from action of group G (fiber of P) of F. a natural question raise here: is this unique way of gluing?is the relation ~ that defined above plays a crucial role in construction of bundle? can we propose another relation that gives same bundle structure? i guess it can be like that: (p,f)~(p.g,g.f) if the first relation is crucial element of construction,how?show me it is unique and not all considerations could be satisfied by another relation. if it is not true,why this way introduces in textbooks without any discussion about general case? and what is general case?